Finally, to sum up, let us highlight the importance of some parts of the present work. Note that the layers of the \(\mathbb {Z}_2\) -extensions of \({k}=\mathbb {Q}(\sqrt{d}, \sqrt{-1})\)  were subject of some recent studies (e.g. [1]}, [2]}); and in this paper, we give more arithmetical properties of \(L_{m,d}\)  (resp. \(L_{m,d}^+\) ), the layers of the cyclotomic \(\mathbb {Z}_2\) -extension of \({k}\)  (resp. \( \mathbb {Q}(\sqrt{d} )\) ).
Furthermore, we discuss the interesting question of the parity of the class number of \(L_{m,d}\) , and we explicitly give the rank of its 2-class group which is strongly related to the interesting problem of the structure of the Iwasawa module (see for example Corollary REF  or [3]}). The authors of [3]} used this paper with some other techniques of Iwasawa theory to determine the structure of the 2-class group of some fields \(L_{m,d}\) .
